Resolving subcategories and dimensions in recollements of extriangulated categories

Abstract

Recently, Wang, Wei and Zhang introduced the notion of recollements of extriangulated categories. In this paper, let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories. We provide some methods to construct resolving subcategories involved in a recollement and study how they are related. As applications of the Auslander–Reiten correspondence, we get the gluing of cotilting modules in a recollement of module categories for artin algebras. We also give some bounds of resolution dimensions of the categories involved in $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ with respect to resolving subcategories, which generalize some known results.

1 Introduction

Abelian categories and triangulated categories are two fundamental structures in algebra and geometry. Nakaoka and Palu [25] introduced the notion of extriangulated categories which is extracting properties on triangulated categories and exact categories (in particular, abelian categories). Many results given in exact categories and triangulated categories can be unified in the same framework (see [9, 12,13,14, 16, 18, 19, 32, 33]). Recollements of triangulated categories and abelian categories were introduced by Beĭlinson et al. [4], which play an important role in algebraic geometry and representation theory. They are closely related to each other and possess similar properties in many aspects. In order to give a simultaneous generalization of recollements of abelian categories and triangulated categories, Wang et al. [27] introduced the notion of recollements of extriangulated categories.

Recently, gluing techniques with respect to a recollement of extriangulated categories have been investigated; for instance, He et al. [11] glued torsion pairs in a recollement of extriangulated categories; Liu and Zhou [20] glued cotorsion pairs in a recollement of extriangulated categories; etc. These results give a simultaneous generalization of recollements of abelian categories and triangulated categories.

Resolving subcategories play an important role in the study of triangulated categories and abelian categories. In abelian categories, resolving subcategories and resolution dimensions are closely related to tilting theory (see [2]) and some homological conjectures (see [17]). Resolving subcategories and resolution dimensions are also crucial to study the relative homological theory in triangulated categories (see [23, 24]). In the situation of recollements, many authors studied the (relative) homological dimension in a recollement of abelian categories or triangulated categories (see [10, 15, 26, 28, 29, 31]).

In this paper, we will provide some methods to construct resolving subcategories and investigate the relation of resolution dimension with respect to resolving subcategories in a recollement of extriangulated categories. The paper is organized as follows:

In Sect. 2, we summarize some basic definitions and properties of extriangulated categories needed in the sequel. In Sect. 3, we first recall the definition of resolving subcategories of extriangulated categories. Latter, we give some methods to construct resolving subcategories via a recollement of extriangulated categories. As applications, using the Auslander–Reiten correspondence [2]), we get the gluing of cotilting modules along with a recollement of module categories for artin algebras. In Sect. 4, we give some bounds for resolution dimensions with respect to resolving subcategories of the categories involved in a recollement of extriangulated categories. Finally, in Sect. 5, we give some examples to illustrate the obtained results.

Throughout this paper, all subcategories are assumed to be full, additive and closed under isomorphisms. Let ${\mathcal {C}}$ be an extriangulated category, and let ${\mathcal {X}}$ be a class of objects of ${\mathcal {C}}$. We use ${\text {add}}{\mathcal {X}}$ to denote the subcategory of ${\mathcal {C}}$ consisting of direct summands of finite direct sums of objects in ${\mathcal {X}}$. Let A be an artin algebra. We use ${\textbf {mod}} \,A$ to denote the category of finitely generated left A-modules.

2 Preliminaries

Let us briefly recall some definitions and basic properties of extriangulated categories from [25]. We omit some details here, but the reader can find them in [25].

Let ${\mathcal {C}}$ be an additive category equipped with an additive bifunctor

$$\begin{aligned} {\mathbb {E}}: {\mathcal {C}}^{\textrm{op}}\times {\mathcal {C}}\rightarrow \textrm{Ab}, \end{aligned}$$

where $\textrm{Ab}$ is the category of abelian groups. For any objects $A, C\in {\mathcal {C}}$, an element $\delta \in {\mathbb {E}}(C,A)$ is called an ${\mathbb {E}}$-extension. Let ${\mathfrak {s}}$ be a correspondence which associates an equivalence class

to any ${\mathbb {E}}$-extension $\delta \in {\mathbb {E}}(C, A)$. This ${\mathfrak {s}}$ is called a realization of ${\mathbb {E}}$, if it makes the diagrams in [25, Definition 2.9] commutative. A triplet $({\mathcal {C}}, {\mathbb {E}}, {\mathfrak {s}})$ is called an extriangulated category if it satisfies the following conditions.

• (1) ${\mathbb {E}}:{\mathcal {C}}^{\textrm{op}}\times {\mathcal {C}}\rightarrow {\textrm{Ab}}$ is an additive bifunctor.

• (2) ${\mathfrak {s}}$ is an additive realization of ${\mathbb {E}}$.

• (3) ${\mathbb {E}}$ and ${\mathfrak {s}}$ satisfy the compatibility conditions in [25, Definition 2.12].

Many examples can be founded in [13, 18, 25].

Example 2.1

(1) Exact categories, triangulated categories and extension-closed subcategories of triangulated categories are extriangulated categories. Extension-closed subcategories of an extriangulated category are also extriangulated categories (see [25, Remark 2.18]).

(2) If ${\mathcal {C}}$ is a triangulated category with suspension functor $\Sigma$ and $\xi$ is a proper class of triangles (see [5] for details). Set ${\mathbb {E}}_{\xi }:=\{\delta \in {\textrm{Hom}}(C, \Sigma A) \mid$ there is a triangle $A \longrightarrow B \longrightarrow C {\mathop {\longrightarrow }\limits ^{\delta }} \Sigma A$ in $\xi \}$ and ${\mathfrak {s}}_{\xi }:=\left. {\mathfrak {s}}\right| _{\xi }$, then $({\mathcal {C}},{\mathbb {E}}_{\xi },{\mathfrak {s}}_{\xi })$ is an extriangulated category (see [13, Remark 3.3]).

We recall the following notations from [25].

• (1) A sequence is called a conflation if it realizes some ${\mathbb {E}}$-extension $\delta \in {\mathbb {E}}(C,A)$. In this case, is called an inflation and is called a deflation. We call an ${\mathbb {E}}$-triangle.

• (2) Let ${\mathcal {X}}$ be a subcategory of ${\mathcal {C}}$, and let be any ${\mathbb {E}}$-triangle.

  • (i) We call A the cocone of and denote it by $\textrm{cocone}(y);$ we call C the cone of and denote it by $\textrm{cone}(x)$.

  • (ii) ${\mathcal {X}}$ is closed under extensions if $A, C\in {\mathcal {X}}$, it holds that $B\in {\mathcal {X}}$.

  • (iii) ${\mathcal {X}}$ is closed under cocones (resp., cones) if $B, C\in {\mathcal {X}}$ (resp., $A,B\in {\mathcal {X}}$), it holds that $A\in {\mathcal {X}}$ (resp., $C\in {\mathcal {X}}$).

Throughout this paper, for an extriangulated category ${\mathcal {C}}$, we assume the following condition, which is analogous to the weak idempotent completeness (see [7, Proposition 7.6]).

Condition 2.2

(WIC) (see [25, Condition 5.8]) Let $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ be any composable pair of morphisms in ${\mathcal {C}}$.

• (1) If gf is an inflation, then f is an inflation.

• (2) If gf is a deflation, then g is a deflation.

Definition 2.3

([25, Definitions 3.23 and 3.25]) Let ${\mathcal {C}}$ be an extriangulated category.

• (1) An object P in ${\mathcal {C}}$ is called projective if for any ${\mathbb {E}}$-triangle $A{\mathop {\longrightarrow }\limits ^{x}}B{\mathop {\longrightarrow }\limits ^{y}}C{\mathop {\dashrightarrow }\limits ^{}}$ and any morphism c in ${\mathcal {C}}(P,C)$, there exists b in ${\mathcal {C}}(P,B)$ such that $yb=c$. We denote the full subcategory of projective objects in ${\mathcal {C}}$ by ${\mathcal {P}}({\mathcal {C}})$. Dually, the injective objects are defined, and the full subcategory of injective objects in ${\mathcal {C}}$ is denoted by ${\mathcal {I}}({\mathcal {C}})$.

• (2) We say that ${\mathcal {C}}$ has enough projectives if for any object $M\in {\mathcal {C}}$, there exists an ${\mathbb {E}}$-triangle $A{\mathop {\longrightarrow }\limits ^{}}P{\mathop {\longrightarrow }\limits ^{}}M{\mathop {\dashrightarrow }\limits ^{}}$ satisfying $P\in {\mathcal {P}}({\mathcal {C}})$. Dually, we define that ${\mathcal {C}}$ has enough injectives.

Remark 2.4

$\mathcal {P(C)}$ is closed under direct summands, extensions and cocones. Dually, $\mathcal {I(C)}$ is closed under direct summands, extensions and cones.

In extriangulated categories, the notions of the left (right) exact sequences (resp., functor) can be founded in [27, Definitions 2.9 and 2.12] for details. Also the notion of extriangulated (resp., exact) functor between two extriangulated categories can be found in [6] (resp., [27, Definition 2.13]).

We also need the following notion.

Definition 2.5

([27, Definition 2.8]) Let ${\mathcal {C}}$ be an extriangulated category. A morphism f in ${\mathcal {C}}$ is called compatible provided that the following condition holds:

f is both an inflation and a deflation implies that f is an isomorphism.

That is, the class of compatible morphisms is the class

$$\begin{aligned} \{f~|~f~\text {is~not~an~inflation},~\text {or}~f~\text {is~not~a~deflation},~\text {or}~f~\text {is~an~isomorphism}\}. \end{aligned}$$

Now we recall the concept of recollements of extriangulated categories [27], which gives a simultaneous generalization of recollements of triangulated categories and abelian categories (see [4, 8]).

Definition 2.6

([27, Definition 3.1]) Let ${\mathcal {A}}$, ${\mathcal {B}}$ and ${\mathcal {C}}$ be three extriangulated categories. A recollement of ${\mathcal {B}}$ relative to ${\mathcal {A}}$ and ${\mathcal {C}}$, denoted by (${\mathcal {A}}$, ${\mathcal {B}}$, ${\mathcal {C}}$), is a diagram

(2.1)

given by two exact functors $i_{*},j^{*}$, two right exact functors $i^{*}$, $j_!$ and two left exact functors $i^{!}$, $j_*$, which satisfies the following conditions:

• (R1) $(i^{*}, i_{*}, i^{!})$ and $(j_!, j^*, j_*)$ are adjoint triples.

• (R2) $\textrm{Im}\,i_{*}={\text {Ker}}j^{*}$.

• (R3) $i_*$, $j_!$ and $j_*$ are fully faithful.

• (R4) For each $B\in {\mathcal {B}}$, there exists a left exact ${\mathbb {E}}$-triangle sequence

  

  in ${\mathcal {B}}$ with $A\in {\mathcal {A}}$, where $\theta _B$ and $\vartheta _B$ are given by the adjunction morphisms.

• (R5) For each $B\in {\mathcal {B}}$, there exists a right exact ${\mathbb {E}}$-triangle sequence

  

  in ${\mathcal {B}}$ with $A'\in {\mathcal {A}}$, where $\upsilon _B$ and $\nu _B$ are given by the adjunction morphisms.

We collect some properties of recollements of extriangulated categories (see [27]).

Lemma 2.7

Let (${\mathcal {A}}$, ${\mathcal {B}}$, ${\mathcal {C}}$) be a recollement of extriangulated categories as in (2.1).

(1) All the natural transformations

$$\begin{aligned} i^{*}i_{*}\Rightarrow \textrm{Id}\,_{{\mathcal {A}}},~\textrm{Id}\,_{{\mathcal {A}}}\Rightarrow i^{!}i_{*},~\textrm{Id}\,_{{\mathcal {C}}}\Rightarrow j^{*}j_{!},~j^{*}j_{*}\Rightarrow \textrm{Id}\,_{{\mathcal {C}}} \end{aligned}$$

are natural isomorphisms. Moreover, $i^{!}$, $i^{*}$ and $j^{*}$ are dense.

(2) $i^{*}j_!=0$ and $i^{!}j_*=0$.

(3) $i^{*}$ preserves projective objects and $i^{!}$ preserves injective objects.

$(3')$ $j_{!}$ preserves projective objects and $j_{*}$ preserves injective objects.

(4) If $i^{!}$ (resp., $j_{*}$) is exact, then $i_{*}$ (resp., $j^{*}$) preserves projective objects.

$(4')$ If $i^{*}$ (resp., $j_{!}$) is exact, then $i_{*}$ (resp., $j^{*}$) preserves injective objects.

(5) If ${\mathcal {B}}$ has enough projectives, then ${\mathcal {A}}$ has enough projectives and $\mathcal {P(A)}={\text {add}}i^{*}(\mathcal {P(B)})$; in addition, if $j^{*}$ preserves projectives, then ${\mathcal {C}}$ has enough projectives and $\mathcal {P(C)}={\text {add}}j^{*}(\mathcal {P(B)})$.

(6) If $i^{!}$ is exact, then $j_{*}$ is exact.

$(6')$ If $i^{*}$ is exact, then $j_{!}$ is exact.

(7) If $i^{!}$ is exact, for each $B\in {\mathcal {B}}$, there is an ${\mathbb {E}}$-triangle

in ${\mathcal {B}}$ where $\theta _B$ and $\vartheta _B$ are given by the adjunction morphisms.

$(7')$ If $i^{*}$ is exact, for each $B\in {\mathcal {B}}$, there is an ${\mathbb {E}}$-triangle

in ${\mathcal {B}}$ where $\upsilon _B$ and $\nu _B$ are given by the adjunction morphisms.

3 Gluing resolving subcategories in a recollement

Throughout this paper, we will always assume that all extriangulated categories admit enough projective objects and injective objects.

Now we recall the notion of resolving subcategory in an extriangulated category, which gives a simultaneous generalization of resolving subcategories in an abelian category (see [1]) and a triangulated category with a proper class $\xi$ of triangles (see [24]).

Definition 3.1

(cf. [32, Page 243]) Let ${\mathcal {C}}$ be an extriangulated category and ${\mathcal {X}}$ a subcategory of ${\mathcal {C}}$. Then, ${\mathcal {X}}$ is called a resolving subcategory of ${\mathcal {C}}$ if the following conditions are satisfied.

• (1) $\mathcal {P(C)}\subseteq {\mathcal {X}}$.

• (2) ${\mathcal {X}}$ is closed under extensions.

• (3) ${\mathcal {X}}$ is closed under cocones.

Clearly, ${\mathcal {C}}$ and $\mathcal {P(C)}$ are trivial resolving subcategories of ${\mathcal {C}}$.

Remark 3.2

• (1) If ${\mathcal {C}}$ is an abelian category, the resolving subcategory defined as above coincide with the earlier one given by Zhu in [34].

• (2) If ${\mathcal {C}}$ is a triangulated category with a proper class $\xi$ of triangles, the resolving subcategory defined as above coincides with the earlier one given by Ma and Zhao in [24]

From now on, let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories. The following result gives a method to glue a resolving subcategory in ${\mathcal {B}}$ from resolving subcategories in ${\mathcal {A}}$ and ${\mathcal {C}}$.

Theorem 3.3

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1). Assume that $\mathcal {X'}$ and $\mathcal {X''}$ are resolving subcategories of ${\mathcal {A}}$ and ${\mathcal {C}}$, respectively. If $i^{*}$ is exact and $j^{*}$ preserves projective objects, then

$$\begin{aligned} {\mathcal {X}}:=\{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {X'}\ \text {and} \ j^{*}(B)\in \mathcal {X''}\} \end{aligned}$$

is a resolving subcategory of ${\mathcal {B}}$. In particular, we have

• (1) $i_{*}(\mathcal {X'})\subseteq {\mathcal {X}}$ and $j_{!}(\mathcal {X''})\subseteq {\mathcal {X}}$.

• (2) $i^{*}({\mathcal {X}})=\mathcal {X'}$ and $j^{*}({\mathcal {X}})=\mathcal {X''}$.

Proof

Since $i^{*}$ preserves projective objects by Lemma 2.7 and $j^{*}$ preserves projective objects by assumption, we have $\mathcal {P(B)}\subseteq {\mathcal {X}}$.

Let be an ${\mathbb {E}}$-triangle in ${\mathcal {B}}$. Applying the exact functors $i^{*}$ and $j^{*}$ to the above ${\mathbb {E}}$-triangle yields the following ${\mathbb {E}}$-triangles

and

If X and Z are in ${\mathcal {X}}$, then $i^{*}(X)\in \mathcal {X'}$, $i^{*}(Z)\in \mathcal {X'}$, $j^{*}(X)\in \mathcal {X''}$, $j^{*}(Z)\in \mathcal {X''}$. Notice that $\mathcal {X'}$ and $\mathcal {X''}$ are closed under extensions, so $i^{*}(Y)\in \mathcal {X'}$ and $j^{*}(Y)\in \mathcal {X''}$, then $Y\in {\mathcal {X}}$ and ${\mathcal {X}}$ is closed under extensions.

If Y and Z are in ${\mathcal {X}}$, then $i^{*}(Y)\in \mathcal {X'}$, $i^{*}(Z)\in \mathcal {X'}$, $j^{*}(Y)\in \mathcal {X''}$, $j^{*}(Z)\in \mathcal {X''}$. Note that $\mathcal {X'}$ and $\mathcal {X''}$ are closed under cocones, we have $i^{*}(X)\in \mathcal {X'}$ and $j^{*}(X)\in \mathcal {X''}$, then $X\in {\mathcal {X}}$ and ${\mathcal {X}}$ is closed under cocones.

Thus, ${\mathcal {X}}$ is a resolving subcategory of ${\mathcal {B}}$.

In particular,

(1) Since $i^{*}i_{*}\cong \textrm{Id}\,_{{\mathcal {A}}}$ by Lemma 2.7 and $j^{*}i_{*}=0$ by assumption, we get $i_{*}(\mathcal {X'})\in {\mathcal {X}}$. Similarly, we have $j_{!}(\mathcal {X''})\in {\mathcal {X}}$.

(2) By (1), we have $\mathcal {X'}\cong i^{*}i_{*}(\mathcal {X'})\subseteq i^{*}({\mathcal {X}})$. Notice that $i^{*}({\mathcal {X}})\subseteq \mathcal {X'}$ is obvious. Thus, $i^{*}({\mathcal {X}})=\mathcal {X'}$. Similarly, we have $j^{*}({\mathcal {X}})=\mathcal {X''}$. $\square$

On the other hand, the following result gives a method to construct resolving subcategories in ${\mathcal {A}}$ and ${\mathcal {C}}$ from a resolving subcategory in ${\mathcal {B}}$.

Theorem 3.4

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1). Assume that ${\mathcal {X}}$ is a resolving subcategory of ${\mathcal {B}}$. Set ${\mathcal {X}}'={\text {add}}i^{*}({\mathcal {X}})$, ${\mathcal {X}}''={\text {add}}j^{*}({\mathcal {X}})$. Then, we have the following statements.

• (1) If $i_{*}({\mathcal {X}}')\subseteq {\mathcal {X}}$, then ${\mathcal {X}}'$ is a resolving subcategory of ${\mathcal {A}}$.

• (2) If $j_{!}({\mathcal {X}}'')\subseteq {\mathcal {X}}$, $j_{!}$ is exact and $j^{*}$ preserves projectives, then ${\mathcal {X}}''$ is a resolving subcategory of ${\mathcal {C}}$.

• (3) If $i_{*}({\mathcal {X}}')\subseteq {\mathcal {X}}$, $j_{!}({\mathcal {X}}'')\subseteq {\mathcal {X}}$ and $i^{*}$ is exact, then

  $$\begin{aligned} {\mathcal {X}}=\{B\in {\mathcal {B}}\mid i^{*}(B)\in {\mathcal {X}}'\ \text {and} \ j^{*}(B)\in {\mathcal {X}}''\}. \end{aligned}$$

Proof

(1) Since $\mathcal {P(B)}\subseteq {\mathcal {X}}$ by assumption, we have $i^{*}(\mathcal {P(B)})\subseteq i^{*}({\mathcal {X}})$. Moreover, by Lemma 2.7, $\mathcal {P(A)}={\text {add}}i^{*}(\mathcal {P(B)})$, thus $\mathcal {P(A)}\subseteq {\mathcal {X}}'$.

Let

be an ${\mathbb {E}}$-triangle in ${\mathcal {A}}$. Applying the exact functor $i_{*}$ to the above ${\mathbb {E}}$-triangle yields the following ${\mathbb {E}}$-triangle

in ${\mathcal {B}}$.

If $X,Z\in {\mathcal {X}}'$, since $i_{*}({\mathcal {X}}')\subseteq {\mathcal {X}}$ by assumption and ${\mathcal {X}}$ is closed under extensions, $i_{*}(Y)\in {\mathcal {X}}$. So $Y\cong i^{*}i_{*}(Y)\in i^{*}({\mathcal {X}})$, and thus $Y\in {\mathcal {X}}'$. Then, ${\mathcal {X}}'$ is closed under extensions.

If $Y,Z\in {\mathcal {X}}'$, since $i_{*}({\mathcal {X}}')\subseteq {\mathcal {X}}$ by assumption and ${\mathcal {X}}$ is closed under cocones, $i_{*}(X)\in {\mathcal {X}}$. So $X\cong i^{*}i_{*}(X)\in i^{*}({\mathcal {X}})$, and thus $X\in {\mathcal {X}}'$. Then, ${\mathcal {X}}'$ is closed under cocones.

Thus, $\mathcal {X'}$ is a resolving subcategory of ${\mathcal {A}}$.

(2) Since $j^{*}$ preserves projectives by assumption, we have $\mathcal {P(C)}={\text {add}}j^{*}(\mathcal {P(B)})$ by Lemma 2.7. Moreover, since $\mathcal {P(B)}\subseteq {\mathcal {X}}$, we have $j^{*}(\mathcal {P(B)})\subseteq j^{*}({\mathcal {X}})$, and hence $\mathcal {P(C)}\subseteq {\mathcal {X}}''$.

Let

be an ${\mathbb {E}}$-triangle in ${\mathcal {C}}$. Applying the exact functor $j_{!}$ to the above ${\mathbb {E}}$-triangle yields the following ${\mathbb {E}}$-triangle:

in ${\mathcal {B}}$.

Assume $X,Z\in {\mathcal {X}}''$. Since $j_{!}({\mathcal {X}}'')\subseteq {\mathcal {X}}$ by assumption and ${\mathcal {X}}$ is closed under extensions, $j_{!}(Y)\in {\mathcal {X}}$. So $Y\cong j^{*}j_{!}(Y) \in j^{*}({\mathcal {X}})$, and hence $Y\in {\mathcal {X}}''$. Then, ${\mathcal {X}}''$ is closed under extensions.

Assume $Y,Z\in {\mathcal {X}}''$. Since $j_{!}({\mathcal {X}}'')\subseteq {\mathcal {X}}$ by assumption and ${\mathcal {X}}$ is closed under cocones, $j_{!}(X)\in {\mathcal {X}}$. So $X\cong j^{*}j_{!}(X) \in j^{*}({\mathcal {X}})$, and hence $X\in {\mathcal {X}}''$. Then, ${\mathcal {X}}''$ is closed under cocones.

Thus, ${\mathcal {X}}''$ is a resolving subcategory of ${\mathcal {C}}$.

(3) It is obvious that

$$\begin{aligned} {\mathcal {X}}\subseteq \{B\in {\mathcal {B}}\mid i^{*}(B)\in {\mathcal {X}}'\ \text {and} \ j^{*}(X)\in {\mathcal {X}}''\}. \end{aligned}$$

We only need to prove the contrary. Let $B\in {\mathcal {B}}$ such that $i^{*}(B)\in {\mathcal {X}}'$ and $j^{*}(B)\in {\mathcal {X}}''$. Since $i^{*}$ is exact by assumption, by Lemma 2.7, there exists an ${\mathbb {E}}$-triangle

in ${\mathcal {B}}$. Note that $i_{*}i^{*}(B)\in i_{*}({\mathcal {X}}')\subseteq {\mathcal {X}}$ and $j_{!}j^{*}(B)\in j_{!}({\mathcal {X}}'')\subseteq {\mathcal {X}}$, we have $B\in {\mathcal {X}}$ by the fact that ${\mathcal {X}}$ is closed under extensions. Thus,

$$\begin{aligned} \{B\in {\mathcal {B}}\mid i^{*}(B)\in {\mathcal {X}}'\ \text {and} \ j^{*}(B)\in {\mathcal {X}}''\}\subseteq {\mathcal {X}} \end{aligned}$$

as desired. $\square$

Also, we have the following result.

Proposition 3.5

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1), and let ${\mathcal {X}}$ be a resolving subcategory of ${\mathcal {B}}$. Set ${\mathcal {X}}''={\text {add}}j^{*}({\mathcal {X}})$. If $j_{*}({\mathcal {X}}'')\subseteq {\mathcal {X}}$ and $j_{*}$ is exact, then ${\mathcal {X}}''$ is a resolving subcategory of ${\mathcal {C}}$.

Proof

Since $j_{*}$ is exact by assumption, $j^{*}$ preserves projectives by Lemma 2.7, and hence $\mathcal {P(C)}={\text {add}}j^{*}(\mathcal {P(B)})$. Observe that $\mathcal {P(B)}\subseteq {\mathcal {X}}$ shows $j^{*}(\mathcal {P(B)})\subseteq j^{*}({\mathcal {X}})$, and thus $\mathcal {P(C)}\subseteq {\mathcal {X}}''$.

Let

be an ${\mathbb {E}}$-triangle in ${\mathcal {C}}$. Applying the exact functor $j_{*}$ to the above ${\mathbb {E}}$-triangle yields the following ${\mathbb {E}}$-triangle

in ${\mathcal {B}}$.

Assume $X,Z\in {\mathcal {X}}''$, since $j_{*}({\mathcal {X}}'')\subseteq {\mathcal {X}}$ by assumption and ${\mathcal {X}}$ is closed under extensions, $j_{*}(Y)\in {\mathcal {X}}$. So $Y\cong j^{*}j_{*}(Y) \in j^{*}({\mathcal {X}})$, and hence $Y\in {\mathcal {X}}''$. Then, ${\mathcal {X}}''$ is closed under extensions.

Assume $Y,Z\in {\mathcal {X}}''$, since $j_{*}({\mathcal {X}}'')\subseteq {\mathcal {X}}$ by assumption and ${\mathcal {X}}$ is closed under cocones, $j_{*}(X)\in {\mathcal {X}}$. So $X\cong j^{*}j_{*}(X) \in j^{*}({\mathcal {X}})$, and hence $X\in {\mathcal {X}}''$. Then, ${\mathcal {X}}''$ is closed under cocones.

Thus, ${\mathcal {X}}''$ is a resolving subcategory of ${\mathcal {C}}$. $\square$

At the end of this section, we give some applications. We recall here the notion of cotilting modules for an artin algebra.

Definition 3.6

([2]) Let A be an artin algebra and $T\in {\textbf {mod}} \,A$. T is called cotilting if it satisfies the following conditions.

• (1) T is self-orthogonal, that is, $\textrm{Ext}\,_{A}^{i}(T,T)=0$ for all $i>0$.

• (2) ${\textrm{id}\,_{A}}T<\infty$.

• (3) For any injective A-module I, there exist some integer n and an exact sequence

  

  in ${\textbf {mod}} \,A$ with $T_{i}\in {\text {add}}T$ for $0\le i\le n$.

For a subclass ${\mathcal {X}}$ of ${\textbf {mod}} \,A$, set

$$\begin{aligned} {^{\perp }{\mathcal {X}}}:=\{M\in {\textbf {mod}} \,A\mid \textrm{Ext}\,_{A}^{i}(M,{\mathcal {X}})=0\text { for all }i>0\}. \end{aligned}$$

Dually, ${\mathcal {X}}^{\perp }$ is defined.

A subcategory ${\mathcal {X}}$ of ${\textbf {mod}} \,A$ is said to be contravariantly finite if for any A-module C, there exists a morphism with $X\in {\mathcal {X}}$ such that is exact ([2, Page 114]).

The following is the well-known Auslander–Reiten correspondence, which classifies basic cotilting modules using contravariantly finite resolving subcategories. Recall from [2, Page 112] that an A-module T is basic if in a direct sum decomposition into indecomposable modules, no indecomposable module appears more than once.

Theorem 3.7

([2, Corollary 5.6]) Let A be an artin algebra of finite global dimension. Then, the assignment $T\longrightarrow {^{\perp }T}$ gives a one-one correspondence between isomorphism classes of basic cotilting A-modules and contravariantly finite resolving subcategories of ${\textbf {mod}} \,A$.

Following the above result, for a contravariantly finite resolving subcategory ${\mathcal {X}}$ of ${\textbf {mod}} \,A$, T is a cotilting A-module satisfying ${\text {add}}T={\mathcal {X}}\cap {\mathcal {X}}^{\perp }$ (see ([2, Theorem 5.5])). In this setting, we say that there exists a cotilting A-module T relative to the subcategory ${\mathcal {X}}:={^{\perp }T}$.

Proposition 3.8

([30, Proposition 3.1]) Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of abelian categories. Assume that $\mathcal {X'}$ and $\mathcal {X''}$ are contravariantly finite subcategories of ${\mathcal {A}}$ and ${\mathcal {C}}$, respectively. If $i^{*}$ is exact, then

$$\begin{aligned} {\mathcal {X}}:=\{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {X'}\ \text {and} \ j^{*}(B)\in \mathcal {X''}\} \end{aligned}$$

is a contravariantly finite subcategory of ${\mathcal {B}}$.

Combining Theorems 3.3 and 3.7, and Proposition 3.8, we can get a gluing method of cotilting modules as follows.

Proposition 3.9

Let A, B and C be artin algebras with finite global dimensions, and let

be a recollement of module categories. Assume that $T'$ and $T''$ are cotilting modules in ${\textbf {mod}} \,A$ and ${\textbf {mod}} \,C$, respectively. If $i^{*}$ is exact and $j^{*}$ preserves projectives, then there exists a cotilting B-module T glued by $T'$ and $T''$, that is, T is a cotilting module relative to the subcategory

$$\begin{aligned} {\mathcal {X}}:=\{B\in {\textbf {mod}} \,B\mid i^{*}(B)\in {^{\perp }T'}\ \text {and} \ j^{*}(B)\in {^{\perp }T''}\} \end{aligned}$$

of ${\textbf {mod}} \,B$ such that ${\mathcal {X}}={^{\perp }T}$.

Remark 3.10

In fact, in Proposition 3.9, one can only assume that A and C have finite global dimensions. In this case, by [26, Theorem 4.4], the global dimension of B is automatically finite.

Conversely, we have

Proposition 3.11

Let A, B and C be artin algebras with finite global dimensions, and let

be a recollement of module categories. Assume that T is a cotilting B-module. Let ${\mathcal {X}}'={\text {add}}i^{*}({^\perp T})$ and ${\mathcal {X}}''={\text {add}}j^{*}({^\perp T})$. Then, we have the following statements.

• (1) If $i_{*}({\mathcal {X}}')\subseteq {^\perp T}$, then there exists a cotilting A-module $T'$ relative to the subcategory ${\mathcal {X}}'$.

• (2) If $j_{!}({\mathcal {X}}'')\subseteq {^\perp T}$, $j_{!}$ is exact and $j^{*}$ preserves projectives, then there exists a cotilting C-module $T''$ relative to the subcategory ${\mathcal {X}}''$.

Proof

Assume that T is a cotilting B-module, then ${^{\perp }T}$ is a contravariantly finite resolving subcategory of ${\textbf {mod}} \,B$ by Theorem 3.7. By [22, Lemma 3.6], ${\mathcal {X}}'$ and ${\mathcal {X}}''$ are contravariantly finite subcategories of ${\textbf {mod}} \,A$ and ${\textbf {mod}} \,C$, respectively. By Theorem 3.4(1), ${\mathcal {X}}'$ is a resolving subcategory of ${\textbf {mod}} \,A$. By Theorem 3.4(2), ${\mathcal {X}}''$ is a resolving subcategory of ${\textbf {mod}} \,C$. Then, the assertions follow from Theorem 3.7. $\square$

Proposition 3.12

Let A, B and C be artin algebras with finite global dimensions, and let

be a recollement of module categories. Assume that T is a cotilting B-module. Let ${\mathcal {X}}''={\text {add}}j^{*}({^\perp T})$. If $j_{*}({\mathcal {X}}'')\subseteq {^\perp T}$ and $j_{*}$ is exact, then there exists a cotilting C-module $T''$ relative to the subcategory ${\mathcal {X}}''$.

Proof

Assume that T is a cotilting B-module, then ${^{\perp }T}$ is a contravariantly finite resolving subcategory of ${\textbf {mod}} \,B$ by Theorem 3.7. By [22, Lemma 3.6], ${\mathcal {X}}''$ is a contravariantly finite subcategories of ${\textbf {mod}} \,C$. By Theorem 3.4(2), ${\mathcal {X}}''$ is a resolving subcategory of ${\textbf {mod}} \,C$. Then, the assertion follows from Theorem 3.7. $\square$

4 Resolving resolution dimensions and recollements

In this section, we give some bounds for resolution dimension of (resolving) subcategories in a recollement of extriangulated categories.

Recall from [32] that an ${\mathbb {E}}$-triangle sequence is defined as a sequence

$$\begin{aligned} \cdots {\longrightarrow }X_{n+2}{\mathop {\longrightarrow }\limits ^{d_{n+2}}}X_{n+1}{\mathop {\longrightarrow }\limits ^{d_{n+1}}} X_{n}{\mathop {\longrightarrow }\limits ^{d_{n}}}X_{n-1}{\longrightarrow }\cdots \end{aligned}$$

in ${\mathcal {C}}$ such that for any n, there exist ${\mathbb {E}}$-triangles $K_{n+1}{\mathop {\longrightarrow }\limits ^{g_{n}}}X_{n}{\mathop {\longrightarrow }\limits ^{f_{n}}}K_{n}{\mathop {\dashrightarrow }\limits ^{}}$ and the differential $d_n=g_{n-1}f_n$.

Now we introduce the notion of resolution dimension for a subcategory of ${\mathcal {C}}$.

Definition 4.1

Let ${\mathcal {X}}$ be a subcategory of ${\mathcal {C}}$ and M an object in ${\mathcal {C}}$. The ${\mathcal {X}}$-resolution dimension of M, written ${\mathcal {X}}$-${\text {res.dim}}M$, is defined by

The resolution dimension of ${\mathcal {C}}$, denoted by ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {C}}$, is defined as

$$\begin{aligned} {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {C}}:=\sup \{{\mathcal {X}}\hbox {-}{\text {res.dim}}M\mid M\in {\mathcal {C}}\}. \end{aligned}$$

In case for ${\mathcal {X}}=\mathcal {P(C)}$, ${\mathcal {X}}\hbox {-}{\text {res.dim}}M$ coincides with ${\text {pd}}M$ (the projective dimension of M), and ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {C}}$ coincides with ${\text {gl.dim}}{\mathcal {C}}$ (the global dimension of ${\mathcal {C}}$) defined in [10].

Now, we give some useful facts.

The following result gives a simultaneous generalization of [34, Lemma 2.2] for an abelian category and [24, Proposition 3.4] for a triangulated category with a proper class $\xi$ of triangles.

Lemma 4.2

Let ${\mathcal {C}}$ be an extriangulated category, and let ${\mathcal {X}}$ be a resolving subcategory of ${\mathcal {C}}$. Assume that

is an ${\mathbb {E}}$-triangle in ${\mathcal {C}}$. Then, we have the following statements.

• (1) ${\mathcal {X}}$-${\text {res.dim}}Y\le \max \{{\mathcal {X}}\hbox {-}{\text {res.dim}}X, {\mathcal {X}}\hbox {-}{\text {res.dim}}Z\}$.

• (2) ${\mathcal {X}}$-${\text {res.dim}}X\le \max \{{\mathcal {X}}\hbox {-}{\text {res.dim}}Y, {\mathcal {X}}\hbox {-}{\text {res.dim}}Z-1\}$.

• (3) ${\mathcal {X}}$-${\text {res.dim}}Z\le \max \{{\mathcal {X}}\hbox {-}{\text {res.dim}}X+1, {\mathcal {X}}\hbox {-}{\text {res.dim}}Y\}$.

Proof

The proof is similar to [24, Proposition 3.4]. $\square$

The following result gives a simultaneous generalization of [34, Lemma 2.1] for an abelian category and [24, Proposition 3.2] for a triangulated category with a proper class $\xi$ of triangles.

Lemma 4.3

Let ${\mathcal {C}}$ be an extriangulated category and ${\mathcal {X}}$ a resolving subcategory of ${\mathcal {C}}$. For any object $M\in {\mathcal {C}}$, if

and

are ${\mathbb {E}}$-triangle sequences with all $X_{i}$ and $Y_{i}$ in ${\mathcal {X}}$ for $0\le i\le n-1$, then $X_{n}\in {\mathcal {X}}$ if and only if $Y_{n}\in {\mathcal {X}}$.

Proof

The proof is similar to [24, Proposition 3.2]. $\square$

Lemma 4.4

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1), and let $\mathcal {X_{B}}$ and $\mathcal {X_{C}}$ be resolving subcategories of ${\mathcal {B}}$ and ${\mathcal {C}}$, respectively. Let $C\in {\mathcal {C}}$. Then,

$$\begin{aligned} \mathcal {X_{B}}\hbox {-} {\text {res.dim}}j_{!}(C) \le \mathcal {X_{C}}\hbox {-}{\text {res.dim}}C+ \max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\} + 1. \end{aligned}$$

Proof

If $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}C=\infty$ or $\max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\}=\infty$, there is nothing to prove. Assume that $\max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\}=m$. The proof will be proceed by induction on the $\mathcal {X_{C}}$-resolution dimension of C. If $C\in \mathcal {X_{C}}$, the assertion holds obviously. Now suppose that $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}C=n\ge 1$. By Lemma 4.3, we have the following ${\mathbb {E}}$-triangle sequence

with $P_{i}\in \mathcal {P(C)}\subseteq \mathcal {X_C}$ for $0\le i\le n-1$ and $X_{n}\in \mathcal {X_C}$.

Notice that $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}K_{1}\le n-1$, by induction hypothesis, we have

$$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(K_{1})\le \mathcal {X_{C}}\hbox {-}{\text {res.dim}}K_{1}+m+1\le n-1+m+1=m+n. \end{aligned}$$

Since $j_{!}$ is right exact, there is an ${\mathbb {E}}$-triangle $K_{1}'{\mathop {\longrightarrow }\limits ^{h_{2}}}j_{!}(P_{0}){\mathop {\longrightarrow }\limits ^{j_{!}(g)}}j_{!}(C){\mathop {\dashrightarrow }\limits ^{}}$ in ${\mathcal {B}}$ and a deflation which is compatible, such that $j_{!}(f)=h_{2}h_{1}$.

Since $j^{*}j_{!}\cong \textrm{Id}_{{\mathcal {C}}}$ by Lemma 2.7,

is an ${\mathbb {E}}$-triangle in ${\mathcal {C}}$. Since $f=j^{*}j_{!}(f)=(j^{*}(h_{2}))(j^{*}(h_{1}))$, so $j^{*}(h_{1})$ is an inflation by Condition 2.2. Note that $j^{*}(h_{1})$ is a deflation and compatible since $j^{*}$ is exact, we have that $j^{*}(h_{1})$ is an isomorphism. Thus, $j^{*}j_{!}(K_{1})\cong j^{*}(K'_{1})$. Set $K''_{1}=\textrm{cocone}(h_{1})$, consider the following ${\mathbb {E}}$-triangle

(4.1)

in ${\mathcal {B}}$. Since $j^{*}$ is exact, $j^{*}(K''_{1})=0$. By (R2), there exists an object $A'\in {\mathcal {A}}$ such that $K''_{1}\cong i_{*}(A')$. Then, $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}K''_{1}\le m$ by assumption. Apply Lemma 4.2 to (4.1), we have $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}K'_{1}\le m+n$. Notice that $j_{!}$ preserves projectives by Lemma 2.7, so $j_{!}(P_{0})\in \mathcal {P(B)}(\subseteq {\mathcal {X}}_{{\mathcal {B}}})$. It follows that $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(C)\le m+n+1$. $\square$

Now we give the main theorem.

Theorem 4.5

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1), and assume that ${\mathcal {B}}$ and ${\mathcal {C}}$ have enough projective objects. Let $\mathcal {X_{A}}$, $\mathcal {X_{B}}$ and $\mathcal {X_{C}}$ be resolving subcategories of ${\mathcal {A}},\ {\mathcal {B}}$ and ${\mathcal {C}}$, respectively. Then, we have the following statements.

• (1) $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}} \le \mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+\max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\} + 1.$

• (2) $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})\le \mathcal {X_{A}}\hbox {-}{\text {res.dim}}{\mathcal {A}}+\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(\mathcal {X_{A}})$.

• (3) $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}} \le \mathcal {X_{A}}\hbox {-}{\text {res.dim}}{\mathcal {A}}+ \mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+\max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(\mathcal {X_{A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\} + 1.$

• (4) $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}\le \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}+ \mathcal {X_{C}}\hbox {-}{\text {res.dim}}j^{*}(\mathcal {X_{B}})$.

• (5) If $j_{!}(\mathcal {X_{C}})\subseteq \mathcal {X_{B}}$ or $j_{*}(\mathcal {X_{C}})\subseteq \mathcal {X_{B}}$, then

  $$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}\le \mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})+1 \end{aligned}$$

  and

  $$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}\le \mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+ \mathcal {X_{A}}\hbox {-}{\text {res.dim}}{\mathcal {A}}+\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(\mathcal {X_A})+1. \end{aligned}$$

• (6) If $i_{*}(\mathcal {X_{A}}) \subseteq \mathcal {X_{B}}$, then $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})\le \mathcal {X_{A}}\hbox {-}{\text {res.dim}}{\mathcal {A}}$.

• (7) If $j_{!}(\mathcal {X_{C}}) \subseteq \mathcal {X_{B}}$ (or $j_{*}(\mathcal {X_{C}}) \subseteq {\mathcal {X_{B}}}$) and $i_{*}(\mathcal {X_{A}}) \subseteq \mathcal {X_{B}}$, then

  $$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}\le \mathcal {X_{A}}\hbox {-}{\text {res.dim}}{\mathcal {A}}+\mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+1. \end{aligned}$$

• (8) If $j^{*}(\mathcal {X_{B}}) \subseteq \mathcal {X_{C}}$, then $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}\le \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}$.

Proof

(1) Suppose that $\max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}),\mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\} = m < \infty$ and $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}= n <\infty$. Let $B\in {\mathcal {B}}$. By (R5), there exists a commutative diagram

in ${\mathcal {B}}$ such that $i_{*}(A'){\mathop {\longrightarrow }\limits ^{}}j_{!}j^{*}(B){\mathop {\longrightarrow }\limits ^{h_{2}}}B'{\mathop {\dashrightarrow }\limits ^{}}$ and $B'{\mathop {\longrightarrow }\limits ^{h_{1}}}B{\longrightarrow }i_{*}i^{*}(B){\mathop {\dashrightarrow }\limits ^{}}$ are ${\mathbb {E}}$-triangles and $h_{2}$ is compatible. Notice that $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(A')\le m$ and $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}i^{*}(B)\le m$. By Lemmas 4.2 and ,

$$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}B&\le \max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}B',\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}i^{*}(B)\}\\&\le \max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(A')+1,\mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}j^{*}(B),\\&\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}i^{*}(B)\} \ \ \ (\textrm{Lemma } 4.2) \\&\le \max \{m+1,\mathcal {X_{C}}\hbox {-}{\text {res.dim}}j^{*}(B)+m+1,m\} \ \ \ \ \ \ (\textrm{Lemma}\ 4.4) \end{aligned}$$

Note that $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}j^{*}(B)\le n$, so $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}B\le m+n+1$.

(2) Suppose that $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(\mathcal {X_{A}})=n<\infty$ and $\mathcal {X_{A}}\hbox {-}{\text {res.dim}}{\mathcal {A}}=m< \infty$. Let $A\in {\mathcal {A}}$. If $A\in \mathcal {X_{A}}$, then $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(A)\le n$ and our result holds. Now suppose that $\mathcal {X_{A}}\hbox {-}{\text {res.dim}}A=s\le m$. Consider the following ${\mathbb {E}}$-triangle sequence

in ${\mathcal {A}}$ with $X'_{i}\in \mathcal {X_{A}}$ for $0\le i\le s$. Since $i_{*}$ is exact,

is an ${\mathbb {E}}$-triangle sequence in ${\mathcal {B}}$. Note that $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(X'_{i})\le n$ by assumption, we have $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(A)\le s+n \le m+n$ by Lemma 4.2.

(3) By (1) and (2), we have

$$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}\le&\mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+\max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\} + 1\\ \le&\mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+\max \{ \mathcal {X_{A}}\hbox {-}{\text {res.dim}}{\mathcal {A}}\\&\quad +\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(\mathcal {X_{A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\} + 1\\ \le&\mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}\\&\quad + \mathcal {X_{A}}\hbox {-}{\text {res.dim}}{\mathcal {A}}+\max \{\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(\mathcal {X_{A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})\} + 1. \end{aligned}$$

(4) Suppose that $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}= m<\infty$ and $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}j^{*}(\mathcal {X_{B}})=n<\infty$. For any object $C\in {\mathcal {C}}$, $j_{!}(C)\in {\mathcal {B}}$. Assume that $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_!(C)=s\le m$, and consider the following ${\mathbb {E}}$-triangle sequence

in ${\mathcal {B}}$ with $X_{i}\in \mathcal {X_B}$ for $0\le i\le s$.

Since $j^{*}$ is exact,

is an ${\mathbb {E}}$-triangle sequence in ${\mathcal {C}}$. Notice that $\mathcal {X_C}\hbox {-}{\text {res.dim}}j^{*}(X_{i})\le n$ by assumption, so $\mathcal {X_C}\hbox {-}{\text {res.dim}}C\le s+n \le m+n$ by Lemma 4.2.

(5) If $j_{!}(\mathcal {X_{C}})\subseteq \mathcal {X_{B}}$, then $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_{C}})=0$. So by (1),

$$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}\le \mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})+1. \end{aligned}$$

If $j_{*}(\mathcal {X_{C}})\subseteq \mathcal {X_{B}}$, for every object $M\in \mathcal {X_{C}}$, $j_{!}(M)\in {\mathcal {B}}$. By (R4), there exists a commutative diagram

in ${\mathcal {B}}$ such that $i_{*}i^{!}(j_{!}(M)){\mathop {\longrightarrow }\limits ^{}}j_{!}(M){\mathop {\longrightarrow }\limits ^{h'_{2}}}B''{\mathop {\dashrightarrow }\limits ^{}}$ and $B''{\mathop {\longrightarrow }\limits ^{h'_{1}}}j_{*}j^{*}(j_{!}(M)){\mathop {\longrightarrow }\limits ^{h'}}i_{*}(A''){\mathop {\dashrightarrow }\limits ^{}}$ are ${\mathbb {E}}$-triangles and $h'_{1}$ is compatible. We note that $j_{*}j^{*}(j_{!}(M))\cong j_{*}(M)\in \mathcal {X_{B}}$ by Lemma 2.7 and assumption, then $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{*}j^{*}(j_{!}(M))=0$. Then, by Lemma 4.2,

$$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(M)&\le \max \{ \mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}B''\}\\&\le \max \{ \mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}), \mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})-1\}\\&\le \mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}}). \end{aligned}$$

So $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}j_{!}(\mathcal {X_C})\le \mathcal {X_B}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})$. Also, by (1), we get

$$\begin{aligned} \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}\le \mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}+\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})+1. \end{aligned}$$

The last assertion follows from (3).

(6) If $i_{*}(\mathcal {X_{A}}) \subseteq \mathcal {X_{B}}$, then $\mathcal {X_{B}}\hbox {-}{\text {res.dim}}i_{*}(\mathcal {X_{A}})=0$. The desired assertion follows from (2).

(7) It follows from (5) and (6).

(8) If $j^{*}(\mathcal {X_{B}})\subseteq \mathcal {X_{C}}$, then $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}j^{*}(\mathcal {X_{B}})=0$. So $\mathcal {X_{C}}\hbox {-}{\text {res.dim}}{\mathcal {C}}\le \mathcal {X_{B}}\hbox {-}{\text {res.dim}}{\mathcal {B}}$ by (4). $\square$

Remark 4.6

One can get the result [29, Theorem 3.7] by applying Theorem 4.7 to abelian categories.

Take $\mathcal {X_{A}}=\mathcal {P(A)}$, $\mathcal {X_{B}}=\mathcal {P(B)}$ and $\mathcal {X_{C}}=\mathcal {P(C)}$ in Theorem 4.5. We have

Corollary 4.7

([10, Theorem 3.5]) Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1), and let $C\in {\mathcal {C}}$. Then, we have the following statements.

• (1)

  • (a) ${{\text {gl.dim}}{\mathcal {B}}}\le \mathcal {P(B)}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})+{\text {gl.dim}}{\mathcal {C}}+1$.

  • (b) $\mathcal {P(B)}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})\le {\text {gl.dim}}{\mathcal {A}}+\sup \{{{\text {pd}}_{{\mathcal {B}}}}i_{*}(P)\mid P\in {\mathcal {P}}({\mathcal {A}})\}$.

  • (c) ${\text {gl.dim}}{\mathcal {B}}\le {\text {gl.dim}}{\mathcal {A}}+{\text {gl.dim}}{\mathcal {C}}+\sup \{{\text {pd}}_{{\mathcal {B}}}i_{*}(P)\mid P\in \mathcal {P(A)}\}+1.$

  • (d) ${\text {gl.dim}}{\mathcal {C}}\le {\text {gl.dim}}{\mathcal {B}}+\sup \{{{\text {pd}}_{{\mathcal {C}}}}j^{*}(P)\mid P\in {\mathcal {P}}({\mathcal {B}})\}$.

• (2) Assume that $i^{!}$ is exact. Then

  • (a) ${\text {gl.dim}}{\mathcal {B}}\le {\text {gl.dim}}{\mathcal {A}}+{\text {gl.dim}}{\mathcal {C}}+1.$

  • (b) ${\text {gl.dim}}{\mathcal {C}}\le {\text {gl.dim}}{\mathcal {B}}.$

Proof

(1) (a) Since $j_{!}(\mathcal {P(C)})\subseteq \mathcal {P(B)}$ by Lemma 2.7, the assertion follows from Theorem 4.5(5).

(b) It follows from Theorem 4.5(2).

(c) It follows from Theorem 4.5(5) (or (a) and (b)).

(d) It follows from Theorem 4.5(4).

(2) (a) Since $i^{!}$ is exact by assumption, $i_{*}(\mathcal {P(A)})\subseteq \mathcal {P(B)}$ by Lemma 2.7. Observe that $j_{!}(\mathcal {P(C)})\subseteq \mathcal {P(B)}$. The assertion follows from Theorem 4.5(7).

(b) Since $i^{!}$ is exact by assumption, $j^{*}(\mathcal {P(B)})\subseteq \mathcal {P(C)}$ by Lemma 2.7. The assertion follows from Theorem 4.5(8). $\square$

Remark 4.8

In the sense of [10], the following dimension

$$\begin{aligned} \mathcal {P_{B}}\hbox {-}{\text {res.dim}}i_{*}({\mathcal {A}})=\sup \{{\text {pd}}_{{\mathcal {B}}}i_{*}(A)\mid A\in {\mathcal {A}}\} \end{aligned}$$

is denoted by ${\text {gl.dim}}_{{\mathcal {A}}}{\mathcal {B}}$.

Under some conditions, one can take special resolving subcategories $\mathcal {X_A}=\mathcal {X'}$, $\mathcal {X_B}={\mathcal {X}}$ and $\mathcal {X_C}=\mathcal {X''}$ of ${\mathcal {A}}$, ${\mathcal {B}}$ and ${\mathcal {C}}$, respectively, as in Theorem 3.3, then we have

Theorem 4.9

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1). Assume that $i^{*}$ is exact and $j^{*}$ preserves projective objects, and assume that $\mathcal {X'}$ and $\mathcal {X''}$ are resolving subcategories of ${\mathcal {A}}$ and ${\mathcal {C}}$ respectively. Then,

$$\begin{aligned} {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}= \max \{\mathcal {X'}\hbox {-}{\text {res.dim}}{\mathcal {A}},\mathcal {X''}\hbox {-}{\text {res.dim}}{\mathcal {C}}\}, \end{aligned}$$

where ${\mathcal {X}}:=\{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {X'}\ \text {and} \ j^{*}(B)\in \mathcal {X''}\}$.

Proof

The inequation $\le$: Assume that $\mathcal {X'}\hbox {-}{\text {res.dim}}{\mathcal {A}}=m$ and $\mathcal {X''}\hbox {-}{\text {res.dim}}{\mathcal {C}}=n$. Let B be an object in ${\mathcal {B}}$. Since $i^{*}$ is exact, by Lemma 2.7, there exists an ${\mathbb {E}}$-triangle

in ${\mathcal {B}}$. Since $i^{*}(B)\in {\mathcal {A}}$ and $j^{*}(B)\in {\mathcal {C}}$, we have $\mathcal {X'}\hbox {-}{\text {res.dim}}i^{*}(B)\le m$ and $\mathcal {X''}\hbox {-}{\text {res.dim}}j^{*}(B)\le n$. Notice that $i_{*}$ is exact and $i_{*}(\mathcal {X'})\subseteq {\mathcal {X}}$ (by Theorem 3.3), so ${\mathcal {X}}\hbox {-}{\text {res.dim}}i_{*}i^{*}(B)\le m$. Since $i^{*}$ is exact, $j_{!}$ is exact by Lemma 2.7. Notice that $j_{!}(\mathcal {X''})\subseteq {\mathcal {X}}$ by Theorem 3.3, so ${\mathcal {X}}\hbox {-}{\text {res.dim}}j_{!}j^{*}(B)\le n$. It follows that ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}\le \max \{m,n\}$ from Lemma 4.2.

The inequation $\ge$: Assume that ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}=l$ and $A\in {\mathcal {A}}$. Since $i_{*}(A)\in {\mathcal {B}}$, we have ${\mathcal {X}}\hbox {-}{\text {res.dim}}i_{*}(A)\le l$. By Theorem 3.3, $i^{*}({\mathcal {X}})\subseteq \mathcal {X'}$. Note that $i^{*}$ is exact by assumption. Then, $\mathcal {X'}\hbox {-}{\text {res.dim}}i^{*}i_{*}(A)\le l$, and thus $\mathcal {X'}\hbox {-}{\text {res.dim}}A\le l$, $\mathcal {X'}\hbox {-}{\text {res.dim}}{\mathcal {A}}\le l$. By Theorem 4.5(8), we have $\mathcal {X''}\hbox {-}{\text {res.dim}}{\mathcal {C}}\le {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}= l$. It follows that $\max \{\mathcal {X'}\hbox {-}{\text {res.dim}}{\mathcal {A}},\mathcal {X''}\hbox {-}{\text {res.dim}}{\mathcal {C}}\}\le {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}.$

Therefore,

$$\begin{aligned} {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}= \max \{\mathcal {X'}\hbox {-}{\text {res.dim}}{\mathcal {A}},\mathcal {X''}\hbox {-}{\text {res.dim}}{\mathcal {C}}\}. \end{aligned}$$

$\square$

In general, $\mathcal {P(B)}\ne \{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {P(A)}\ \text {and} \ j^{*}(B)\in \mathcal {P(C)}\}$ (see Example 5.1(3)), but we have the following result.

Lemma 4.10

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1). We have the following statements.

• (1) If $j^{*}$ preserves projectives, then

  $$\begin{aligned} \mathcal {P(B)}\subseteq \{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {P(A)}\ \text {and} \ j^{*}(B)\in \mathcal {P(C)}\}. \end{aligned}$$

• (2) If $i^{*}$ and $i^{!}$ are exact, then

  $$\begin{aligned} \mathcal {P(B)}=\{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {P(A)}\ \text {and} \ j^{*}(B)\in \mathcal {P(C)}\}. \end{aligned}$$

Proof

(1) By Lemma 2.7, we have $i^{*}(\mathcal {P(B)})\subseteq \mathcal {P(A)}$. Since $j^{*}$ preserves projectives, $j^{*}(\mathcal {P(B)})\subseteq \mathcal {P(C)}$. Then,

$$\begin{aligned} \mathcal {P(B)}\subseteq \{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {P(A)}\ \text {and} \ j^{*}(B)\in \mathcal {P(C)}\}. \end{aligned}$$

(2) Since $i^{!}$ is exact by assumption, $j^{*}$ preserves projectives by Lemma 2.7. Then, by (1), we have

$$\begin{aligned} \mathcal {P(B)}\subseteq \{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {P(A)}\ \text {and} \ j^{*}(B)\in \mathcal {P(C)}\}. \end{aligned}$$

Conversely, let $B\in {\mathcal {B}}$ such that $i^{*}(B)\in \mathcal {P(A)}$ and $j^{*}(B)\in \mathcal {P(C)}$. Since $i^{*}$ is exact, there exists an ${\mathbb {E}}$-triangle

(4.2)

in ${\mathcal {B}}$.

Since $i^{!}$ is exact, $i_{*}$ preserves projectives by Lemma 2.7. Notice that $j_{!}$ preserves projectives by Lemma 2.7, so $j_{!}j^{*}(B)$ and $i_{*}i^{*}(B)$ are projective objects in ${\mathcal {B}}$. By Remark 2.4, we have $B\in \mathcal {P(B)}$. Then, $\{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {P(A)}\ \text {and} \ j^{*}(B)\in \mathcal {P(C)}\}\subseteq \mathcal {P(B)}$, and the desired assertion is obtained. $\square$

Remark 4.11

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1). Assume that $i^{*}$ is exact and $j^{*}$ preserves projectives. One can easily get

$$\begin{aligned} \mathcal {P(B)}\subseteq \{B\in {\mathcal {B}}\mid i^{*}(B)\in \mathcal {P(A)}\ \text {and} \ j^{*}(B)\in \mathcal {P(C)}\} \end{aligned}$$

by Theorem 3.3.

Combining Theorem 4.9 with Lemma 4.10, we have the following result.

Corollary 4.12

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1). If $i^{*}$ and $i^{!}$ are exact, then

$$\begin{aligned} {\text {gl.dim}}{\mathcal {B}}=\max \{{\text {gl.dim}}{\mathcal {A}},{\text {gl.dim}}{\mathcal {C}}\}. \end{aligned}$$

Remark 4.13

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of abelian categories. If $i^{*}$ and $i^{!}$ are exact, then ${\mathcal {B}}\cong {\mathcal {A}} \times {\mathcal {C}}$ (see [8, Corollary 8.10]). Thus, the assertion ${\text {gl.dim}}{\mathcal {B}}=\max \{{\text {gl.dim}}{\mathcal {A}},{\text {gl.dim}}{\mathcal {C}}\}$ is obvious.

Conversely, we have the following result, which holds true for any subcategory ${\mathcal {X}}$ of ${\mathcal {B}}$ (${\mathcal {X}}$ is not necessary a resolving subcategory).

Theorem 4.14

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1). Assume that ${\mathcal {X}}$ is a subcategory of ${\mathcal {B}}$. Then, we have the following statements.

• (1) If $i^{*}$ is exact, then $i^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\mathcal {A}}\le {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}$.

• (2) $j^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\mathcal {C}}\le {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}$.

Proof

Assume that ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}=l$.

(1) Let $A\in {\mathcal {A}}$. Since $i_{*}(A)\in {\mathcal {B}}$, we have ${\mathcal {X}}\hbox {-}{\text {res.dim}}i_{*}(A)\le l$. Note that $i^{*}$ is exact by assumption and $i^{*}i_{*}\cong \textrm{Id}\,_{{\mathcal {A}}}$ by Lemma 2.7, so we have $i^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}i^{*}i_{*}(A)\le l$, thus $i^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}A\le l$ and $i^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\mathcal {A}}\le l$, that is, $i^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\mathcal {A}}\le {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}$.

(2) Let $C\in {\mathcal {C}}$. Since $j_{*}(C)\in {\mathcal {B}}$, we have ${\mathcal {X}}\hbox {-}{\text {res.dim}}j_{*}(C)\le l$. Notice that $j^{*}$ is exact and $j^{*}j_{*}\cong \textrm{Id}\,_{{\mathcal {C}}}$ by Lemma 2.7, we have $j^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}j^{*}j_{*}(C)\le l$, thus $j^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}C\le l$ and $j^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\mathcal {C}}\le l$, that is, $j^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\mathcal {C}}\le {\mathcal {X}}\hbox {-}{\text {res.dim}}{\mathcal {B}}$. $\square$

Taking ${\mathcal {X}}=\mathcal {P(B)}$ in Theorem 4.14, then we have

Corollary 4.15

Let $({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$ be a recollement of extriangulated categories as the diagram (2.1). Then, we have the following statements.

• (1) If $i^{*}$ is exact, ${\text {gl.dim}}{\mathcal {A}}\le {\text {gl.dim}}{\mathcal {B}}$.

• (2) (cf. Corollary 4.7(2)(b)) If $i^{!}$ or $j_{*}$ is exact, then ${\text {gl.dim}}{\mathcal {C}}\le {\text {gl.dim}}{\mathcal {B}}$.

Proof

(1) Notice that $i^{*}(\mathcal {P(B)})\subseteq \mathcal {P(A)}$ by Lemma 2.7, so the desired result follows from Theorem 4.14.

(2) Note that if $i^{!}$ or $j_{*}$ is exact, then $j^{*}(\mathcal {P(B)})\subseteq \mathcal {P(C)}$ by Lemma 2.7, so the desired result follows from Theorem 4.14. $\square$

5 Examples

We give some examples to illustrate the obtained results.

Let A, B be artin algebras and $_{A}M_{B}$ an (A, B)-bimodule, and let $\Lambda ={A\ {M}\atopwithdelims ()\ 0\ \ B}$ be a triangular matrix algebra. Then, any module in ${\textbf {mod}} \,\Lambda$ can be uniquely written as a triple ${X\atopwithdelims ()Y}_{f}$ with $X\in {\textbf {mod}} \,A$, $Y\in {\textbf {mod}} \,B$ and $f\in \textrm{Hom}_{A}(M\otimes _{B}Y,X)$ (see [3, p.76] for more details).

Example 5.1

Let A be a finite-dimensional algebra given by the quiver and B be a finite-dimensional algebra given by the quiver with the relation $\beta \alpha =0$. Define a triangular matrix algebra $\Lambda ={A\ \ A\atopwithdelims ()0\ \ B}$, where the right B-module structure on A is induced by the unique algebra surjective homomorphism satisfying $\phi (e_{3})=e_{1}$, $\phi (e_{4})=e_{2}$, $\phi (e_{5})=0$. Then, $\Lambda$ is a finite-dimensional algebra given by the quiver

with the relations $\gamma \alpha =\delta \epsilon$ and $\beta \alpha =0$. The Auslander–Reiten quiver of $\Lambda$ is

Following [26, Example 2.12], we have that

is a recollement of module categories, where

$$\begin{aligned}&i^{*}\left( {X\atopwithdelims ()Y}_{f}\right) =Y,{} & {} i_{*}(Y)={0\atopwithdelims ()Y}&i^{!}\left( {X\atopwithdelims ()Y}_{f}\right) ={\text {Ker}}\left( Y\rightarrow \textrm{Hom}_{A}(A,X)\right) ,\\&j_{!}(X)={X\atopwithdelims ()0},{} & {} j^{*}\left( {X\atopwithdelims ()Y}_{f}\right) =X,&j_{*}(X)={X\atopwithdelims ()\textrm{Hom}_{A}(A,X)}. \end{aligned}$$

By [21, Lemma 3.2], we know that $i^{*}$ admits a left adjoint $\widetilde{i^{*}}$ and $j_{!}$ admits a left adjoint $\widetilde{j_{!}}$, where

$$\begin{aligned} \widetilde{i^{*}}(Y)={A\otimes _{B} Y\atopwithdelims ()Y}_{1} ,{} & {} \widetilde{j_{!}}({X\atopwithdelims ()Y}_{f})={\text {Coker}}f, \end{aligned}$$

So $i^{*}$ and $j_{!}$ are exact. Since $_{A}A$ is projective, $j_{*}$ is exact.

• (1) Take resolving subcategories

  $$\begin{aligned} \mathcal {X'}&={\mathcal {P}}({\textbf {mod}} \,B)={\text {add}}(P(5)\oplus P(4)\oplus P(3)),\\ {\mathcal {X}}&={\text {add}}\left( {0\atopwithdelims ()P(5)} \oplus {S(2)\atopwithdelims ()0} \oplus {S(2)\atopwithdelims ()P(4)}\oplus {P(1)\atopwithdelims ()0} \oplus {S(2)\atopwithdelims ()S(4)} \oplus {P(1)\atopwithdelims ()P(4)} \oplus {0\atopwithdelims ()P(4)}\oplus {P(1)\atopwithdelims ()S(4)}\oplus {P(1)\atopwithdelims ()P(3)}\right) ,\\ \mathcal {X''}&={\mathcal {P}}({\textbf {mod}} \,A)={\text {add}}(S(2)\oplus P(1)) \end{aligned}$$

  of ${\textbf {mod}} \,B$, ${\textbf {mod}} \,\Lambda$ and ${\textbf {mod}} \,A$ respectively. Clearly $\mathcal {X''}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A=1$ and ${\mathcal {X}}\hbox {-}{\text {res.dim}}i_{*}({\textbf {mod}} \,B)=1$. Then, by Theorem 4.5 (or [29, Theorem 3.7]), we have the following assertions.

  • (1.1) Notice that $j_{!}(\mathcal {X''})= {\text {add}}({S(2)\atopwithdelims ()0} \oplus {P(1)\atopwithdelims ()0})\subseteq {\mathcal {X}}$. Thus, ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda \le \mathcal {X''}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A+{\mathcal {X}}\hbox {-}{\text {res.dim}}i_{*}({\textbf {mod}} \,B)+1=1+1+1=3$.

  • (1.2) Notice that $j^{*}({\mathcal {X}})= {\text {add}}(S(2) \oplus P(1))\subseteq \mathcal {X''}$. Thus, $1=\mathcal {X''}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A\le {\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda$.

  Then, $1\le {\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda \le 3$. In fact, ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda =2$.

• (2) Take resolving subcategories

  $$\begin{aligned} \mathcal {X'}&={\text {add}}(P(5)\oplus P(4)\oplus S(4)\oplus P(3)),\\ {\mathcal {X}}&={\text {add}}\left( {0\atopwithdelims ()P(5)}\oplus {S(2)\atopwithdelims ()P(4)}\oplus {S(2)\atopwithdelims ()0} \oplus {P(1)\atopwithdelims ()0}\oplus {S(2)\atopwithdelims ()S(4)}\oplus {P(1)\atopwithdelims ()P(4)}\oplus \right. \\&~~~~~~~\left. {P(1)\atopwithdelims ()S(4)}\oplus {0\atopwithdelims ()P(4)}\oplus {S(1)\atopwithdelims ()0}\oplus {P(1)\atopwithdelims ()P(3)}\oplus {0\atopwithdelims ()S(4)}\oplus {S(1)\atopwithdelims ()P(3)}\oplus {0\atopwithdelims ()P(3)}\right) ,\\ \mathcal {X''}&={\text {add}}(S(2)\oplus P(1)) \end{aligned}$$

  of ${\textbf {mod}} \,B$, ${\textbf {mod}} \,\Lambda$ and ${\textbf {mod}} \,A$ respectively. Clearly, $\mathcal {X'}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,B=2$ and $\mathcal {X''}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A=1$. Then, by Theorem 4.5 (or [29, Theorem 3.7]), we have the following assertions.

  • (2.1) Notice that $i_{*}(\mathcal {X'})= {\text {add}}({0\atopwithdelims ()P(5)} \oplus {0\atopwithdelims ()P(4)}\oplus {0\atopwithdelims ()S(4)}\oplus {0\atopwithdelims ()P(3)})\subseteq {\mathcal {X}}$. So ${\mathcal {X}}\hbox {-}{\text {res.dim}}i_{*}({\textbf {mod}} \,B) \le \mathcal {X'}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,B=2$. It is clear that ${\mathcal {X}}\hbox {-}{\text {res.dim}}i_{*}({\textbf {mod}} \,B) =1$.

  • (2.2) Notice that $j_{!}(\mathcal {X''})= {\text {add}}({S(2)\atopwithdelims ()0} \oplus {P(1)\atopwithdelims ()0})\subseteq {\mathcal {X}}$. So ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda \le \mathcal {X''}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A +{\mathcal {X}}\hbox {-}{\text {res.dim}}i_{*}({\textbf {mod}} \,A)+1=1+1+1=3$.

  In fact, ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda =1$.

• (3) Take resolving subcategories

  $$\begin{aligned} \mathcal {X'}= & {} {\mathcal {P}}({\textbf {mod}} \,B)={\text {add}}(P(5)\oplus P(4)\oplus P(3)), \\ \mathcal {X''}= & {} {\mathcal {P}}({\textbf {mod}} \,A)={\text {add}}(S(2)\oplus P(1)) \end{aligned}$$

  of ${\textbf {mod}} \,B$ and ${\textbf {mod}} \,A$ respectively. Then, by Theorem 3.3, we get a resolving subcategory

  $$\begin{aligned} {\mathcal {X}}&={\text {add}}\left( {0\atopwithdelims ()P(5)}\oplus {S(2)\atopwithdelims ()P(4)}\oplus {S(2)\atopwithdelims ()0} \oplus \right. \\&~~~~~~~\left. {P(1)\atopwithdelims ()0}\oplus {P(1)\atopwithdelims ()P(4)}\oplus {0\atopwithdelims ()P(4)}\oplus {P(1)\atopwithdelims ()P(3)}\oplus {0\atopwithdelims ()P(3)}\right) \end{aligned}$$

  in ${\textbf {mod}} \,\Lambda$. It is obvious that ${\mathcal {P}}({\textbf {mod}} \,\Lambda ) \subsetneqq {\mathcal {X}}$.

  Clearly, $\mathcal {X'}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,B=2$ and $\mathcal {X''}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A=1$. So by Theorem 4.9, ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda =2$.

• (4) Take resolving subcategories

  $$\begin{aligned} \mathcal {X'}= & {} {\text {add}}(P(5)\oplus P(4)\oplus S(4)\oplus P(3)), \\ \mathcal {X''}= & {} {\text {add}}(S(2)\oplus P(1)) \end{aligned}$$

  of ${\textbf {mod}} \,B$ and ${\textbf {mod}} \,A$ respectively. Then, by Theorem 3.3, we get a resolving subcategory

  $$\begin{aligned} {\mathcal {X}}={\text {add}}\left( {0\atopwithdelims ()P(5)}\oplus {S(2)\atopwithdelims ()P(4)}\oplus {S(2)\atopwithdelims ()0} \oplus {P(1)\atopwithdelims ()0}\oplus {S(2)\atopwithdelims ()S(4)}\oplus {P(1)\atopwithdelims ()P(4)}\oplus \right. \\ ~~~~~~~~~~~~~~~~~~~~~~\left. {P(1)\atopwithdelims ()S(4)}\oplus {0\atopwithdelims ()P(4)}\oplus {P(1)\atopwithdelims ()P(3)}\oplus {0\atopwithdelims ()S(4)}\oplus {0\atopwithdelims ()P(3)}\right) \end{aligned}$$

  in ${\textbf {mod}} \,\Lambda$. Clearly, $\mathcal {X'}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,B=1$ and $\mathcal {X''}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A=1$. So by Theorem 4.9, ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda =1$.

• (5) Take a resolving subcategory

  $$\begin{aligned} {\mathcal {X}}={\text {add}}\left( {0\atopwithdelims ()P(5)}\oplus {S(2)\atopwithdelims ()P(4)}\oplus {S(2)\atopwithdelims ()0} \oplus {P(1)\atopwithdelims ()0}\oplus {S(2)\atopwithdelims ()S(4)}\oplus {P(1)\atopwithdelims ()P(4)}\oplus \right. \\ ~~~~~~~\left. {P(1)\atopwithdelims ()S(4)}\oplus {0\atopwithdelims ()P(4)}\oplus {S(1)\atopwithdelims ()0}\oplus {P(1)\atopwithdelims ()P(3)}\oplus {0\atopwithdelims ()S(4)}\oplus {S(1)\atopwithdelims ()P(3)}\oplus {0\atopwithdelims ()P(3)}\right) \end{aligned}$$

  in ${\textbf {mod}} \,\Lambda$. Notice that $i_{*}i^{*}({\mathcal {X}})={\text {add}}({0\atopwithdelims ()P(5)}\oplus {0\atopwithdelims ()P(4)}\oplus {0\atopwithdelims ()S(4)}\oplus {0\atopwithdelims ()P(3)})\subseteq {\mathcal {X}}$ and $j_{!}j^{*}({\mathcal {X}})={\text {add}}({S(1)\atopwithdelims ()0}\oplus {P(1)\atopwithdelims ()0}\oplus {S(2)\atopwithdelims ()0})\subseteq {\mathcal {X}}$. Then, by Theorem 3.4, we have that

  $$\begin{aligned} i^{*}({\mathcal {X}})={\text {add}}(P(5) \oplus P(4)\oplus S(4)\oplus P(3)) \end{aligned}$$

  is a resolving subcategory in ${\textbf {mod}} \,B$, and

  $$\begin{aligned} j^{*}({\mathcal {X}})={\text {add}}(S(2) \oplus P(1)\oplus S(1))={\textbf {mod}} \,A \end{aligned}$$

  is a resolving subcategory in ${\textbf {mod}} \,A$.

  Clearly, ${\mathcal {X}}\hbox {-}{\text {res.dim}}{\textbf {mod}} \,\Lambda =1$. By Theorem 4.14, we have $i^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}} {\textbf {mod}} \,B\le 1$ and $j^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A\le 1$. In fact, $i^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\textbf {mod}} \,B=1$ and $j^{*}({\mathcal {X}})\hbox {-}{\text {res.dim}}{\textbf {mod}} \,A=0$.

• (6) The condition “$i_{*}i^{*}({\mathcal {X}})\subseteq {\mathcal {X}}$” is not necessary in Theorem 3.4(1). Take a resolving subcategory

  $$\begin{aligned} {\mathcal {X}}= & {} {\text {add}}\left( {0\atopwithdelims ()P(5)} \oplus {S(2)\atopwithdelims ()0} \oplus {S(2)\atopwithdelims ()P(4)}\oplus {P(1)\atopwithdelims ()0} \oplus {S(2)\atopwithdelims ()S(4)} \oplus {P(1)\atopwithdelims ()P(4)} \oplus \right. \\{} & {} \left. {0\atopwithdelims ()P(4)}\oplus {P(1)\atopwithdelims ()P(3)} \right) \end{aligned}$$

  in ${\textbf {mod}} \,\Lambda$. Notice that

  $$\begin{aligned} i^{*}({\mathcal {X}})={\text {add}}(P(5) \oplus P(4)\oplus S(4)\oplus P(3)) \end{aligned}$$

  is a resolving subcategory in ${\textbf {mod}} \,B$. But

  $$\begin{aligned} i_{*}i^{*}({\mathcal {X}})={\text {add}}\left( {0\atopwithdelims ()P(5)}\oplus {0\atopwithdelims ()P(4)}\oplus {0\atopwithdelims ()S(4)}\oplus {0\atopwithdelims ()P(3)}\right) \nsubseteq {\mathcal {X}}. \end{aligned}$$